What's the truth about this? How should these claims be answered or addressed?

Basically, the article in question points out distinctions between early Celtic Christianity and Latin Christianity, but these distinctions are not doctrinal, but rather involve liturgical norms and Church disciplines. The article states that such distinctions indicate that the Celtic Church had more in keeping with Eastern Orthodox Churches. Apparently the author concludes that Celtic Christianity was more of a kind of Orthodox Church and not really a part of the Catholic Church.

Here are two points that refute the author's position:

1) Using this same line of reasoning, a person could conclude that the Byzantine Rite Catholics are not really a part of the Catholic Church. After all, there is a big distinction between the Byzantine Rites and the Latin Rite in terms of liturgical norms and Church disciplines (i.e., the same sorts of things the author points out). But, of course, suggesting that Byzantine Rite Catholics are not truly a part of the Catholic Church would be absurd. Therefore, I can look at the points the author made and simply say that they illustrate the distinctions between an early Celtic Rite and the Latin Rite, but they were still rites within the same Church.

2) The big mistake of the author is presuming that Irish Celtic Christianity was confined to the island of Ireland (and therefore all the Irish Christians were subjected to the actions of King Henry II). This was hardly the case. Early Irish Celtic Christians heavily engaged in missionary activity throughout Europe and other parts of the world. The Christian communities they founded were Catholic, not some kind of separate Celtic Orthodox Church. The author gives a theory about how the English put an end to the alleged "Celtic Orthodox Church" in Ireland (making it conform to Latin Catholicism), but how does he explain the non-existence of this "Celtic Orthodox Church" in all the hundreds of Christian communities founded by Irish missionaries throughout the known world? Did Henry II invade all those other places too? Or did all the various world leaders get on board with the mission of Henry II, without a single instance of one of these communities resisting it and preserving their original "Celtic Orthodox Church"?

The simplest explanation why there is no "Celtic Orthodox Church" today is because there never was one to begin with. The early Irish Celtic Church was still part of the Catholic Church. Whatever Henry II did in Ireland, it was not transforming Celtic Christianity from an independent Church into the Catholic Church.